Problem: Simplify; express your answer in exponential form. Assume $k\neq 0, y\neq 0$. $\dfrac{{(k^{5}y^{2})^{3}}}{{(k^{5}y^{3})^{3}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(k^{5}y^{2})^{3} = (k^{5})^{3}(y^{2})^{3}}$ On the left, we have ${k^{5}}$ to the exponent ${3}$ . Now ${5 \times 3 = 15}$ , so ${(k^{5})^{3} = k^{15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(k^{5}y^{2})^{3}}}{{(k^{5}y^{3})^{3}}} = \dfrac{{k^{15}y^{6}}}{{k^{15}y^{9}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{15}y^{6}}}{{k^{15}y^{9}}} = \dfrac{{k^{15}}}{{k^{15}}} \cdot \dfrac{{y^{6}}}{{y^{9}}} = k^{{15} - {15}} \cdot y^{{6} - {9}} = y^{-3}$